JAMES BERNOULLI. 75 



132. James Bernoulli seems to be on more familiar terms 

 with infinity than mathematicians of the present day. On his 

 page 262 we find him stating, correctly, that the sum of the infinite 



series —-r + —p^+ -77, + -77 + . . . is infinite, for the series is greater 

 \/i v^ V^ V"* 



1111 



than 7 + Q + Q + 7 + ... He adds that the sum of all the odd 



terms of the first series is to the sum of all the even terms as 

 \/2 — 1 is to 1 ; so that the sum of the odd terms would appear to 

 be less than the sum of the even terms, which is impossible. But 

 the paradox does not disturb James Bernoulli, for he adds, 



...cujus evavTLO(fiaveLas rationem, etsi ex infiniti natiira finito intel- 

 lectui comprehendi non posse videatur, nos tamen satis perspectam 

 habemus. 



183. At the end of the volume containing the Ars Conjectandi 

 we have the Lettre a un Amy, sur les Parties da Jen de Faume, 

 to which we have alluded in Art. 97. 



The nature of the problem discussed may be thus stated. 

 Suppose A and B two players ; let them play a set of games, say 

 five, that is to say, the player gains the set who first wins five 

 games. Then a certain number of sets, say four, make a match. 

 It is required to estimate the chances of A and B in various states 

 of the contest. Suppose for example that A has won two sets, 

 and B has won one set ; and that in the set now current A has 

 won two games and B has won one game. The problem is thus 

 somewhat similar in character to the Problem of Points, but more 

 complicated. James Bernoulli discusses it very fully, and presents 

 his result in the form of tables. He considers the case in which the 

 players are of unequal skill ; and he solves various problems arising 

 from particular circumstances connected with the game of tennis 

 to which the letter is specially devoted. 



On the second page of the letter is a very distinct statement 

 of the use of the celebrated theorem known by the name of Ber- 

 noulli ; see Art. 123. 



134. One problem occurs in ihi^ Lettre a un Amy... which 

 it may be interesting to notice. 



Suppose that A and B engage in play, and that each in turn 



